Two months after the launch of the state’s new unemployment benefits computer system, it remains plagued with problems, $6 million over initial estimates, and unable to make proper payments to hundreds of financially strapped workers hunting for jobs.

The company that built the system, Deloitte Consulting, has until the end of the month to fix the problems without penalty, under the $46 million contract. It’s unclear what remedies are available to the state if the system is still not working properly after that.

Massachusetts officials declined to answer questions about the specific terms of the contract. But Michelle Amante, the state official in charge of overseeing Deloitte’s rollout, defended the system in an interview this week.

“We fundamentally believe the system is working,” said Amante, a former employee of Deloitte, a New York-based consulting firm. “When we pay out $40-$42 million [in claims] a week, the system is working.”

It certainly doesn’t seem that way to the many jobless Massachusetts residents — from laid-off accountants to out-of-work school bus drivers — who have gone without some or all of their unemployment benefits since the system went into operation July 1. In some cases, they have been erroneously billed for thousands of dollars in alleged overpayments. One Lawrence man said he was mistakenly charged $45,339.

“I’m scared, especially with winter coming up,” said Brittney Reitano of Easton, whose benefits were inexplicably cut to $190 a week from $570. “I’m being penalized because the system doesn’t work.”

After she filed her weekly claim in July, Reitano said, the new system began garnishing her checks to cover back payments of nearly $2,000 it said she owed. Reitano estimated the system so far has mistakenly grabbed more than $4,000 from her benefits, but she can’t find a way to make the deductions stop.

She spent 2 hours and 42 minutes on hold Tuesday afternoon in another vain attempt to reach someone who could resolve her problem. Meanwhile, she is living on dwindling savings to pay her mortgage and help care for her elderly parents — with heating bills fast approaching.

State labor officials said the problems are related to the conversion of data from the old system to the new, affecting a relatively small number of people. About 115,000 people have received benefits claims every week since the system launched July 1, officials said.

To handle the flood of issues, the state extended the hours of call centers, increased overtime, and hired temporary workers. Before the system launched, call centers averaged about 25,000 calls a week; they averaged more than 29,000 in July, the month the system launched, and nearly 26,000 in August, according to the state.

Deloitte has also deployed additional workers to fix the system.

“There is no question this has been a complex and challenging project,” Deloitte said in a statement. “That is why we have invested considerable time and firm resources into making sure UI Online is a quality system that meets the needs of the Commonwealth and the people it serves.’’

Helen O’Donnell of Harvard was laid off from an accounting job in December. After the new system was rolled out, she received 26 notices telling her she owed the state $10,000 — more benefits than she has received, she said.

She has spent hours battling her way through the phone system, waiting on hold as long as 90 minutes, only to be disconnected before reaching someone. She said she finally received a payment this week — after calling elected officials and the governor’s office.

“I was worried about myself financially,” she said. “Some people are worried about their kids.”

State Senator Dan Wolf, a Harwich Democrat who cochairs the Legislature’s Joint Committee on Labor and Workforce Development, said it’s unclear whether the problems are temporary bugs that often accompany new software or something more serious. Either way, he said, the expectation is that Deloitte will fix the problems.

“There really should be no higher priority,” said Wolf, a candidate for governor. “We’re dealing with people who have lost their jobs and are out of work.’’

Jesse Mermell, a spokesman for Governor Deval Patrick, said the administration will ensure that Deloitte meets its contractual obligations. “While most claimants have accessed the new system without problems,” Mermell said, “those who have encountered an issue are rightfully frustrated, and so are we.”

The cost of the computer project has grown from $40 million in 2007 to $46 million, state officials said. The online system for filing unemployment claims is the second phase of a technology upgrade launched under former secretary of Labor and Workforce Development Suzanne Bump, who is now the state auditor.

On her campaign website, Bump lists securing funding “to upgrade ancient and inadequate computer and telephone systems” of the unemployment agency as one of her accomplishments during her nearly three years leading the state labor department.

But in an interview this week, Bump distanced herself from the contract, saying she had nothing to do with the latest rollout and played a relatively small role in the first phase, which got underway during her tenure and launched the month she left to run for state auditor.

“I left in order to start a campaign for auditor. I didn’t keep up with what was going on there,” Bump said. “I was there to ensure that progress was made, that there was employer testing that was done before the launch, and then I left.”

That launch of the first phase, which brought the employer side of the system online, was also rocky as employers experienced problems accessing and using the online system. The state eventually waived fees and deferred interest on delayed unemployment insurance tax payments as a result of the glitches.

Deloitte took over the state contract in May 2009 after buying the public services practice of BearingPoint, the original contractor on the project, which filed for bankruptcy protection earlier that year.

Amante, who is overseeing the system launch as director of the Department of Unemployment Assistance, began working for BearingPoint in 2005. From 2009 to 2010, then with Deloitte, she worked on the design of the state’s new unemployment benefits system, according to an interview and an “appearance of conflict of interest” disclosure form filed with the state.

Joanne Goldstein, appointed by the governor in 2010 to replace Bump as labor secretary, hired Amante from Deloitte as revenue director of the Department of Unemployment Assistance, promoted her to acting director of the department last year, and named her director in February.

“I have confidence Deloitte is getting no special treatment or breaks,” Goldstein said. “I have confidence in her ability to manage the [unemployment insurance] modernization process and keep Deloitte accountable.”

Michael Krigsman, a software analyst and author of the blog “Beyond IT Failure,” said Amante’s role could present the appearance of a conflict of interest. While Amante may bring the benefits of knowing the system from the inside, Krigsman said, “it is hard to go back to your former colleagues and hold their feet to the fire.”

Amante said she did not work at Deloitte with anyone on the current Deloitte management team. About 20 other Deloitte staffers, she said, have remained on the project from her time at the company.

“I think that my background only benefits the state,” Amante said. “I’m able to watch out for things and take care of things in ways other people might not be able to.”